This paper is devoted to the Cauchy problems for the one-dimensional linear time-fractional diffusion equations with ∂ α t the Caputo fractional derivative of order α ∈ (0, 1) in the variable t and time-degenerate diffusive coefficients t β with β > −α. The solutions of Cauchy problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t, are shown. In the "Problem statement and main results"section of the paper, the solution of the time-fractional degenerate diffusion equation in a variable coefficient with two different initial conditions are considered. In this work, a solution is found by using the Kilbas-Saigo function E α,m,l (z) and applying the Fourier transform F and inverse Fourier transform F −1 . Convergence of solution of problem 1 and problem 2 are proven using Plancherel theorem. The existence and uniqueness of the solution of the problem are confirmed.